Escape rate and diffusion of a random walker

We determine the rate of escape from a potential well, and the diffusion coefficient in a periodic potential, of a random walker that moves under the influence of the potential in between successive collisions with the heat bath. In the overdamped limit, both the escape rate and the diffusion coefficient coincide with those of a Langevin particle. Conversely, in the underdamped limit the two dynamics have a different temperature dependence. In particular, at low temperature the random walk has a smaller escape rate, but a larger diffusion coefficient.